Surrounding the solution of a linear system of equations from all sides

Suppose $A \in \mathbb {R}^{n \times n}$ is invertible and we are looking for the solution of $Ax = b$. Given an initial guess $x_1 \in \mathbb {R}$, we show that by reflecting through hyperplanes generated by the rows of $A$, we can generate an infinite sequence $(x_k)_{k=1}^{\infty }$ such that all elements have the same distance to the solution $x$, i.e. $\|x_k - x\| = \|x_1 - x\|$. If the hyperplanes are chosen at random, averages over the sequence converge and \begin{equation*} \mathbb {E} \left \| x - \frac {1}{m} \sum _{k=1}^{m}{ x_k} \right \| \leq \frac {1 + \|A\|_F \|A^{-1}\|}{\sqrt {m}} \cdot \|x-x_1\|. \end{equation*}

The bound does not depend on the dimension of the matrix. This introduces a purely geometric way of attacking the problem: are there fast ways of estimating the location of the center of a sphere from knowing many points on the sphere? Our convergence rate (coinciding with that of the Random Kaczmarz method) comes from simple averaging, can one do better?